Semiconductor
Materials
Department of Electronic and Electrical Engineering
Lecturers: Prof. Tatiana Perova, SNIAM building,
perovat@tcd.ie
OBJECTIVES:
This course deals with an introduction to semiconductor materials.
SYLLABUS:
Semiconductors: Intrinsic silicon, extrinsic n and p type silicon,
mobility of carriers, carrier transport in semiconductors;
p-n junctions.
Semiconductor Materials
•
•
•
•
•
The Semiconductor Industry
Semiconductor devices such as diodes, transistors and integrated circuits
can be found everywhere in our daily lives, in Walkman, televisions,
automobiles, washing machines and computers. We have come to rely on
them and increasingly have come to expect higher performance at lower
cost.
Personal computers clearly illustrate this trend. Anyone who wants to
replace a three to five year old computer finds that the trade-in value of
his (or her) computer is surprising low. On the bright side, one finds that
the complexity and performance of the today’s personal computers vastly
exceeds that of their old computer and that for about the same purchase
price, adjusted for inflation.
While this economic reality reflects the massive growth of the industry, it
is hard to even imagine a similar growth in any other industry. For instance,
in the automobile industry, no one would even expect a five times faster car
with a five times larger capacity at the same price when comparing to what
was offered five years ago. Nevertheless, when it comes to personal
computers, such expectations are very realistic.
The essential fact which has driven the successful growth of the computer
industry is that through industrial skill and technological advances one
manages to make smaller and smaller transistors. These devices deliver year
after year better performance while consuming less power and because of
their smaller size they can also be manufactured at a lower cost per device.
Introduction to Semiconductors
Objective of the lecture:
1. Define a semiconductor – no. of electrons in
outer shell, location on periodic table, most
commonly used ones etc.
2. Know the crystal structure of silicon, the cause
and result of defects.
3. Understand intrinsic and extrinsic
semiconductor behaviour, know how to affect
this behaviour through doping.
4. Explain in detail what depletion regions are and
how they are formed.
5. P-N junction
Why semiconductors?
• SEMICONDUCTORS: They are here, there, and everywhere
• Computers, palm pilots,
Silicon (Si) MOSFETs, ICs, CMOS
laptops, anything “intelligent”
• Cell phones, pagers
Si ICs, GaAs FETs, BJTs
• CD players
AlGaAs and InGaP laser diodes, Si photodiodes
• TV remotes, mobile terminals
Light emitting diodes (LEDs)
• Satellite dishes
InGaAs MMICs (Monolithic Microwave
ICs)
• Fiber networks
InGaAsP laser diodes, pin photodiodes
• Traffic signals, car
GaN LEDs (green, blue)
taillights
InGaAsP LEDs (red, amber)
• Air bags
Si MEMs, Si ICs
• and, they are important, especially to Elec.Eng.& Computer
Sciences
Introduction
Semiconductors are materials whose electrical
properties lie between Conductors and Insulators.
Ex : Silicon and Germanium
Give the examples of Conductors and Insulators!
Difference in conductivity
Semiconductor Materials
• Elemental semiconductors – Si and Ge (column IV of periodic
table) –compose of single species of atoms
• Compound semiconductors – combinations of atoms of column
III and column V and some atoms from column II and VI.
(combination of two atoms results in binary compounds)
• There are also three-element (ternary) compounds (GaAsP) and
four-elements (quaternary) compounds such as InGaAsP.
Semiconductor
materials
Semiconductor Materials
• The wide variety of electronic and optical properties of these
semiconductors provides the device engineer with great
flexibility in the design of electronic and opto-electronic
functions.
• Ge was widely used in the early days of semiconductor
development for transistors and diods.
• Si is now used for the majority of rectifiers, transistors and
integrated circuits.
• Compounds are widely used in high-speed devices and devices
requiring the emission or absorption of light.
• The electronic and optical properties of semiconductors are
strongly affected by impurities, which may be added in precisely
controlled amounts (e.g. an impurity concentration of one part
per million can change a sample of Si from a poor conductor to a
good conductor of electric current). This process called doping.
Solid state structures
A crystalline solid is distinguished by the fact that atoms making the
crystal are arranged in a periodic fashion. That is, there is some basic
arrangement of atoms that is repeated throughout the entire solid.
Thus the crystal appears exactly the same at one point as it does at a
series of other equivalent points, once the basic periodicity is
discovered. However, not all solids are crystals (Fig. 2); some have no
periodic structure at all (amorphous solids), and other are composed
of many small regions of single-crystal material (polycrystalline
solids).
The periodic arrangement of atoms in crystal is called the lattice; the
lattice contains a volume, called a unit cell, which is representative of the
entire lattice and is regularly repeated throughout the crystal.
Solid state structures
• Cubic lattices:
Simple cubic (sc)
Unit cells for types of cubic lattice structure.
Body-centered cubic (bcc)
Face-centered cubic (fcc)
Diamond lattice unit cell, showing the four nearest neighbour structure
The basic lattice structure for many important
semiconductors is the diamond lattice, which is
characteristic of Si and Ge. In many compound
semiconductors, atoms are arranged in a basic
diamond structure but are different on alternating
sites. This is called a zincblende lattice and is
typical of the III-V compounds. The diamond
lattice can be thought of as an fcc structure with
an extra atom placed at a/4+b/4+c/4 from each of
the fcc atoms.
Solid state structures
The diamond lattice of silicon and germanium.
The zinc-blende crystal structure of GaAs and InP
Each atom in the diamond
lattice has a covalent bond
with four adjacent atoms,
which
together
form
a
tetrahedron. This lattice can
also be formed from two fcccubic lattices, which are
displaced along the body
diagonal of the larger cube in
Figure by one quarter of that
body diagonal. The diamond
lattice therefore is a fcc-cubic
lattice with a basis containing
two identical atoms.
Atoms and electrons
We shall investigate some of the important
properties of electrons, with special emphasis on
two points: (1) the electronic structure of atoms
and (2) the interaction of atoms and electrons
with excitation, such as the absorption and
emission of light. By studying electron energies
in an atom, we lay the foundation for
understanding the influence of the lattice on
electrons participating in current flow through a
solid.
One of the most valuable experiments of modern physics is the
analysis of absorption and emission of light by atoms. For
example, an electric discharge can be created in a gas, so that
the atoms begin to emit light with wavelengths characteristic of
the gas.
The result of emission spectra experiments led Niels Bohr
to construct a model for the hydrogen atom, based on
the mathematics of planetary systems. If the electron in
the hydrogen atom has a series of planetary-type orbits
available to it, it can be excited to an outer orbit and then
can fall to any one of the inner orbits, giving off energy
corresponding to one of the lines seen in a spectrum.
The Bohr model
To develop the model, Bohr made several
postulates:
1. Electrons exist in certain stable, circular
orbits about the nucleus.
2. The electron may shift to an orbit of
higher or lower energy, thereby gaining or
losing energy equal to the difference in
the energy levels (by absorption or
emission of a photon of energy hν).
However, the simple Bohr model, which accurately
described the gross features of the hydrogen
spectrum, did not include many fine features. These
features were described later by principles of quantum
mechanics.
The Silicon Atom
Finally, the work of Bohr, Boltzmann, Plank,
Einstein and others has developed an
understanding of the atomic structure which
shows that electrons circle the nucleus in orbits
having different associated energies. The
electrons also spin on their own axes. The energy
of electrons is quantised in that only certain
discrete levels of energy can be possessed by
electrons and no values in between these discrete
levels are allowed. The levels exist in groups
which are referred to as shells and there are
sub-shells (l) within main shells (n).
Silicon, Si, is a group IV material having an atomic number of 14. Consequently it has 14
positively charged protons and 14 neutrons in its nucleus. It has 14 orbiting negatively
charged electrons: 2 in a full K shell; 8 in a full L shell and 4 in a half-full M sub-shell.
With a half full outer sub-shell the atom has an affinity for 4 additional electrons to try to
complete the outer sub-shell.
The Pauli’s Exclusion Principle
states that no two electrons in an atom or molecule can share the exact same quantum
specification. In practice, this means that no more than two electrons can share
precisely the same orbit or energy level and the two must have opposite spins.
The Silicon Atom
A covalent bond can be formed between two atoms
which have only one electron in an outer orbit or
energy level. In this case the individual electrons
from the separate atoms at the same energy level
orbit both atoms jointly as shown in figures.
Both atoms essentially share the pair of electrons at the given energy level in the outer
sub-shell, with the two electrons having opposite spins. This forms a bonding
attraction between the two atoms which is not extremely strong but is nonetheless
powerful and maintains a high degree of stability in the material.
In the case of Silicon, each of the 4 outer electrons enters into a covalent bond
with a neighbouring atom.
A Covalent Bond Formed by the
Sharing of Electrons in an Outer
Energy Level
The Silicon Atomic Structure
-
-
-
-
-
-
-
Si
-
14
Silicon: our primary example and
focus
Atomic no. 14
14 electrons in three shells: 2 ) 8 ) 4
i.e., 4 electrons in the outer "bonding"
shell
Silicon forms strong covalent bonds with
4 neighbors
-
-
-
However, like all
other elements it
would prefer to have
8 electrons in its
outer shell
Band theory of a solid
• A solid is formed by bringing together isolated single atoms.
• Consider the combination of two atoms. If the atoms are far
apart there is no interaction between them and the energy
levels are the same for each atom. The numbers of levels at a
particular energy is simply doubled
n=3
n=3
n=2
n=2
n=1
n=1
Atom 1
Atom 2
• If the atoms are close together the electron wave functions will
overlap and the energy levels are shifted with respect to each
other.
n=3
n=3
n=3
n=2
n=2
n=2
n=1
n=1
n=1
Atom 1
Atom 2
Atom 1 + 2
•
•
•
A solid will have millions of atoms
close together in a lattice so
these energy levels will creates
bands each separated by a gap.
Conductors:
– If we have used up all the
electrons available and a band
is still only half filled, the solid
is said to be a good conductor.
The half filled band is known
as the conduction band.
Insulators:
– If, when we have used up all
the electrons the highest band
is full and the next one is
empty with a large gap
between the two bands, the
material is said to be a good
insulator. The highest filled
band is known as the valence
band while the empty next
band is known as the
conduction band.
n=3
n=2
n=1
Conduction band,
half filled with
electrons
Valence band,
filled with
electrons
Empty
conduction band
Large energy gap
Valence band,
filled with
electrons
Semiconductors:
• Some materials have a filled valence band
just like insulators but a small gap to the
conduction band.
• At zero Kelvin the material behave just
like an insulator but at room temperature,
it is possible for some electrons to
acquire the energy to jump up to the
conduction band. The electrons move
easily through this conduction band under
the application of an electric field. This is
an intrinsic semiconductor.
Empty
conduction band
Small energy gap
Valence bands,
filled with
electrons
At zero Kelvin – no conduction
Conduction
band, with some
electrons
Top valence
band now
missing some
electrons
At room temperature – some conduction
So where are all these materials
to be found in the periodic table ?
Semiconductor
materials
Possible Semiconductor Materials
Carbon
C
6
1. Very Expensive
2. Band Gap Large: 6eV
3. Difficult to produce without high contamination
Si
1. Cheap
14 2. Ultra High Purity
3. Oxide is amazingly perfect for IC applications
Germanium
Ge
1. High Mobility
32 2. High Purity Material
3. Oxide is porous to water/hydrogen (problematic)
Tin
Sn
50
1. Only “White Tin” is semiconductor
2. Converts to metallic form under moderate heat
Lead
Pb
82
1. Only “White Lead” is semiconductor
2. Converts to metallic form under moderate heat
Silicon
Brief introduction to Semiconductors
(conductivity for Si depends on doping, Cu ~ 6E7 -1m-1)
Think of a crystal matrix of silicon atoms (Si has 4 valence
electrons).
Diamond lattice structure
The diamond lattice can be thought of as an fcc structure
with an extra atom placed at a/4+b/4+c/4 from each of the
fcc atoms.
Diamond lattice - http://en.wikipedia.org/wiki/File:Diamond_cubic_animation.gif
The Silicon Atomic Structure
-
-
-
-
-
-
-
Si
-
14
-
-
-
-
-
Silicon : It’s a Group 4 element which means it has 4 electrons
in outer shell
However, like all other elements it would prefer to have 8
electrons in its outer shell
The Germanium Atomic Structure
Bonding of Si atoms
This results in the covalent bonding of Si atoms in the crystal
matrix
A Covalent Bond Formed by the Sharing of
Electrons in an Outer Energy Level
Band Gap Energy
n=3
n=3
n=2
n=2
n=1
n=1
Atom 1
Atom 2
Discrete energy levels for 2 atoms separated by a
large distance.
Typical continuous band pictures at
0 K for different solid materials.
Note that the band gap energy,
Eg for insulators is ~ 10 eV,
while for metals it is close to 0
eV (1eV=1.6x10-19 J).
Electrons and Holes
Si and Ge are tetravalent elements – each atom of Si (Ge) has 4 valence
electrons in crystal matrix
T=0 all electrons are bound in
covalent bonds
no carriers available for
conduction.
For T> 0 thermal fluctuations can
break electrons free creating
electron-hole pairs
Both can move throughout the lattice
and therefore conduct current.
Electrons and Holes
For T>0
some electrons in the valence band receive
enough thermal energy to be excited
across the band gap to the conduction
band.
The result is a material with some electrons
in an otherwise empty conduction band and
some unoccupied states in an otherwise
filled valence band.
An empty state in the valence band is
referred to as a hole.
Electron-hole pairs in a semiconductor.
The bottom of the conduction band
denotes as Ec and the top of the valence
band denotes as Ev.
If the conduction band electron and the
hole are created by the excitation of a
valence band electron to the conduction
band, they are called an electron-hole
pair (EHP).
Silicon Lattice Structure
At 0K, all
electrons
are tightly
shared with
neighbours
 no
current
flow
-
-
Si
-
-
-
Si
-
-
--
-
Si
-
-
Si
-
-
Si
--
Si
--
Si
--
Si
-
Si
-
-
-
--
-
Si
-
Free electron
-
-
-
Si
-
-
-+- Si - - - - - Si - - Si - - Si - --+- Si -
--
Si
--
Adding heat (even to room
temperature) allows some bonds to
break, and electrons can flow
Si
Vacancy
left by
electron.
- Overall
Sicharge
- on
- silicon is
zero 
this “hole”
must be
positive
Shares electrons
with 4 neighbouring
atoms  8
electrons in outer
shell
Intrinsic Material
A perfect semiconductor crystal with no impurities or lattice defects is called an
intrinsic semiconductor.
At T=0 K –
No charge carriers
Valence band is filled with electrons
Conduction band is empty
At T>0
Electron-hole pairs are generated
EHPs are the only charge carriers in
intrinsic material
Since electron and holes are created in
pairs – the electron concentration in
conduction band, n (electron/cm3) is
equal to the concentration of holes in the
valence band, p (holes/cm3).
Each of these intrinsic carrier
concentrations is denoted ni.
Thus for intrinsic materials n=p=ni
Electron-hole pairs in the covalent bonding
model in the Si crystal.
Intrinsic Material
•
•
At a given temperature there is a certain concentration of electron-hole
pairs ni. If a steady state carrier concentration is maintained, there must be
recombination of EHPs at the same rate at which they are generated.
Recombination occurs when an electron in the conduction band makes a
transition to an empty state (hole) in the valence band, thus annihilating the
pair. If we denote the generation rate of EHPs as gi (EHP/cm3·s) and the
recombination rate as ri, equilibrium requires that
ri = gi
Each of these rates is temperature dependent. For example, gi(T) increases
when the temperature is raised, and a new carrier concentration ni is
established such that the higher recombination rate ri (T) just balances
generation. At any temperature, we can predict that the rate of
recombination of electrons and holes ri, is proportional to the equilibrium
concentration of electrons n0 and the concentration of holes p0:
ri = r n0 p0 = r ni2 = gi
•
The factor r is a constant of proportionality which depends on the
particular mechanism by which recombination takes place.
Increasing conductivity by temperature
As temperature increases, the number of free electrons and
holes created increases exponentially.
Carrier Concentration vs Temp (in Si)
17
1 10
16
1 10
15
1 10
14
1 10
Intrinsic Concentration (cm^-3)
13
1 10
12
1 10
11
1 10
ni 1 1010
T
9
1 10
8
1 10
7
1 10
6
1 10
5
1 10
4
1 10
3
1 10
100
150
200
250
300
350
400
450
500
T
Temperature (K)
Therefore the conductivity of a semiconductor is influenced by
temperature
Increasing conductivity
• The conductivity of the semiconductor material increases when the
temperature increases.
• This is because the application of heat makes it possible for some
electrons in the valence band to move to the conduction band.
• Obviously the more heat applied the higher the number of electrons
that can gain the required energy to make the conduction band
transition and become available as charge carriers.
• This is how temperature affects the carrier concentration.
• Another way to increase the number of charge carriers is to add
them in from an external source.
• Doping or implant is the term given to a process whereby one
element is injected with atoms of another element in order to
change its properties.
• Semiconductors (Si or Ge) are typically doped with elements such as
Boron, Arsenic and Phosphorous to change and enhance their
electrical properties.
Semiconductor
materials
Extrinsic Material
By doping, a crystal can be altered so that it has a predominance of either
electrons or holes. Thus there are two types of doped semiconductors, n-type
(mostly electrons) and p-type (mostly holes). When a crystal is doped such that
the equilibrium carrier concentrations n0 and po are different from the intrinsic
carrier concentration ni, the material is said to be extrinsic.
Donor impurities (elements
of group V): P, Sb, As
Acceptor elements (group
III): B, Al, Ga, In
When impurities or lattice
defects are introduced,
additional levels are created
in the energy bands
structure, usually within the
band gap.
Total number of electrons
III – Al – 13
The valence and conduction bands of
silicon with additional impurity energy
levels within the energy gap.
IV – Si – 14
V-
P - 15
Extrinsic Material – donation of electrons
n-type material
Donation of electrons from
a donor level to the
conduction band
An impurity from column V introduces an
energy level very near the conduction band
in Ge or Si. This level is filled with electrons
at 0 K, and very little thermal energy is
required to excite these electrons to the
conduction band. Thus, at about 50-100 K
nearly all of the electrons in the impurity
level are "donated" to the conduction band.
Such an impurity level is called a donor level,
and the column V impurities in Ge or Si are
called donor impurities. From figure we note
that the material doped with donor
impurities can have a considerable
concentration of electrons in the conduction
band, even when the temperature is too low
for the intrinsic EHP concentration to be
appreciable. Thus semiconductors doped
with a significant number of donor atoms
will have n0>>(ni,p0) at room temperature.
This is n-type material.
Extrinsic Material – acceptance of electrons
P-type material
Acceptance of valence band
electrons by an acceptor level,
and the resulting creation of
holes.
Atoms from column III (B, Al, Ga, and
In) introduce impurity levels in Ge or Si
near the valence band. These levels are
empty of electrons at 0 K. At low
temperatures, enough thermal energy is
available to excite electrons from the
valence band into the impurity level,
leaving behind holes in the valence band.
Since this type of impurity level
"accepts" electrons from the valence
band, it is called an acceptor level, and
the column III impurities are acceptor
impurities in Ge and Si. As figure
indicates, doping with acceptor
impurities can create a semiconductor
with a hole concentration p0 much
greater than the conduction band
electron concentration n0 (this is ptype material).
Donor and acceptors in covalent bonding model
Donor and acceptor atoms
in the covalent bonding
model of a Si crystal.
In the covalent bonding model, donor and
acceptor atoms can be visualized as shown in
the Figure. An Sb atom (column V) in the Si
lattice has the four necessary valence
electrons to complete the covalent bonds
with the neighboring Si atoms, plus one extra
electron. This fifth electron does not fit into
the bonding structure of the lattice and is
therefore loosely bound to the Sb atom. A
small amount of thermal energy enables this
extra electron to overcome its coulombic
binding to the impurity atom and be donated
to the lattice as a whole. Thus it is free to
participate in current conduction. This
process is a qualitative model of the
excitation of electrons out of a donor level
and into the conduction band.
Similarly, the column III impurity Al has only
three valence electrons to contribute to the
covalent bonding, thereby leaving one bond
incomplete. With a small amount of thermal
energy, this incomplete bond can be
transferred to other atoms as the bonding
electrons exchange positions.
Increasing conductivity by doping
-
-
Si
-
--
-
Si
-
Si
-+--
-
--Si - As
--Si - As
Si -
-
Si
-
Si
-
- + -+- Si - - Si - -- - Si Si
-Si - - Si - Si
- As
- - Si - - Si - - Si -+• Inject Arsenic into the crystal with an implant step.
- Si • Arsenic is Group5 element
- with 5 electrons in its outer shell, (one more than
silicon).
• This introduces extra electrons into the lattice which can be released through
the application of heat and so produces and electron current
• The result here is an N-type semiconductor (n for negative current carrier)
Increasing conductivity by doping
- Si - - Si -+- Si - Si - + - Si - - Si - - Si -+- Si - - Si - + - - +-+
B - - Si - - Si
B - Si - - Si - - Si
- + - - -+
B - - Si -+- Si - Si - - Si
- Si • Inject Boron into the crystal with an implant step.
• Boron is Group3 element is has 3 electrons in its outer shell (one less than silicon)
• This introduces holes into the lattice which can be made mobile by applying heat. This
gives us a hole current
• The result is a P-type semiconductor (p for positive current carrier)
Calculation of binding energy
We can calculate rather simply the approximate energy required to
excite the fifth electron of a donor atom into the conduction
band (the donor binding energy) based on the Bohr model
results:
mn*q 4
E
2 K 2 2
where m n* is the effective mass typical of semiconductors (mn*  0.12m0
m0 = 9.11x10-31 kg is the electronic rest mass),
is a
reduced Planck’s constant and
K  40 r
where εr is the relative dielectric constant of the semiconductor
material and 0 = 8.85x10-12 F/m is the permittivity of free
space.
The Fermi level
Electrons in solids obey Fermi - Dirac statistics:
1
f (E) 
1  exp ( E  E F ) / kT 
(4.6)
The following consideration
are used in the development
of this statistics:
1. indistinguishability of the
electrons,
where k is Boltzmann’s constant
2. electron wave nature,
k=8.62ּ10-5 eV/K=1.38 10-23 J/K.
3. the Pauli exclusion principle.
The function f(E) called the Fermi-Dirac distribution function gives the probability that
an available energy state at E will be occupied by an electron at absolute temperature T.
The quantity EF is called the Fermi level, and it represents an important quantity in the
analysis of semiconductor behavior. For an energy E = EF the occupation probability is
f ( E F )  1  exp ( E F  E F ) / kT 
1
1
1


11 2
This is the probability for electrons to occupy the Fermi level.
(4.7)
The Fermi – Dirac distribution function
At T=0K f(E) has rectangular shape
the denominator of the exponent is
1/(1+0)=1 when (E<Ef), exp. negative
1/(1+)-0 when (E>Ef), exp. positive
At 0 К every available energy state up to EF is filled
with electrons, and all states above EF are empty.
The Fermi – Dirac distribution
function for different temperatures
At temperatures higher than 0 K, some probability
f(E) exists for states above the Fermi level to be
filled with electrons and there is a corresponding
probability [1 - f(E)] that states below EF are empty.
The Fermi function is symmetrical about EF for all
temperatures. The probability exists for
state E above EF is filled –
state E below EF is filled –
f(EF+ E)
[1- f(EF - E)]
The symmetry of the distribution of empty and filled states about EF makes the Fermi level a
natural reference point in calculations of electron and hole concentrations in semiconductors.
In applying the Fermi-Dirac distribution to semiconductors, we must recall that f(E) is the
probability of occupancy of an available state at E. Thus if there is no available state at E (e.g.,
in the band gap of a semiconductor), there is no possibility of finding an electron there.
Relation between f(E) and the band structure
Electron probability
tail f(E)
Hole probability
tail [1-f(E)]
In intrinsic material the Fermi level EF must lie at the middle of the band gap.
In n-type material the distribution function f(E) must lie above its intrinsic position on the energy scale. The energy
difference (Ec – EF) gives a measure of n.
For p-type material the Fermi level lies near the valence band such that the [1-f(E)] tail below Ev is larger than the
f(E) tail above Ec. The value of (EF – Ev) indicates how strongly p-type the material is.
The distribution function has values within the band gap between Eν and Ec, but there are no energy states
available, and no electron occupancy results from f(E) in this range.
Electron and Hole Concentrations at Equilibrium
The Fermi distribution function can be used to calculate the concentrations of electrons
and holes in a semiconductor if the densities of available states in the valence and
conduction bands are known. The concentration of electrons in the conduction band is

n0 
 f ( E ) N ( E )dE
(4.8)
Ec
where N(E)dE is the density of states (cm-3) in the energy range dE. The subscript 0 used
for the electron and hole concentration symbols (n0, p0) indicates equilibrium conditions.
The number of electrons per unit volume in the energy range dE is the product of the density of
states and the probability of occupancy f(E). Thus the total electron concentration is the integral
over the entire conduction band. The function N(E) can be calculated by using quantum
mechanics and the Pauli exclusion principle.
N(E) is proportional to E1/2, so the density of states in the conduction band increases with
electron energy. On the other hand, the Fermi function becomes extremely small for large
energies. The result is that the product f(E)N(E) decreases rapidly above Ec, and very few
electrons occupy energy states far above the conduction band edge.
Similarly, the probability of finding an empty state (hole) in the valence band [1 - f(E)]
decreases rapidly below Ev, and most holes occupy states near the top of the valence band.
Band diagram, density of states, Fermi-Dirac distribution,
and the carrier concentrations at thermal equilibrium
Intrinsic
semiconductor
n-type
semiconductor
p-type
semiconductor
The conduction band electron concentration is simply the effective density of states at Ec
times the probability of occupancy at Ec:
(4-9)
n0  N c f ( E c )
In this expression we assume the Fermi level EF lies at least several kT below the
conduction band. Then the exponential term is large compared with unity, and the
Fermi function f(Ec) can be simplified as
f ( Ec ) 
1
 exp  ( Ec  EF ) / kT 
1  exp ( Ec  EF ) / kT 
(4-10)
Since kT at room temperature is only 0.026 eV, this is generally a good approximation.
For this condition the concentration of electrons in the conduction band is
n0  N c exp  ( Ec  EF ) / kT 
(4-11)
It can be shown that the effective density of states Nc is
2mn* kT 3 / 2
N c  2(
)
2
h
(4-12)
Values of Nc can be tabulated as a function of temperature. As Eq. (4-11) indicates, the
electron concentration increases as EF moves closer to the conduction band.
By similar arguments, the concentration of holes in the valence band is
p0  N v [1  f ( Ev )]
where Nv is the effective density of states in the valence band.
(4-13)
The probability of finding an empty state at Ev, is
1  f ( Ev )  1 
1
 exp  ( E F  Ev ) / kT 
1  exp ( Ev  E F ) / kT 
(4-14)
for EF larger than Ev by several kT. From these equations, the concentration of holes
in the valence band is
p0  N v exp  ( EF  Ev ) / kT
(4-15)


The effective density of states in the valence band reduced to the band edge is
N v  2(
2m*p kT
h
2
)3 / 2
(4-16)
Eq. (4-15) predicts that the hole concentration increases as EF moves closer to the
valence band.
The electron and hole concentrations predicted by Eqs. (4-11) and (4-15) are valid
whether the material is intrinsic or doped, provided thermal equilibrium is maintained.
Thus for intrinsic material, EF lies at some intrinsic level Ei near the middle of the band
gap, and the intrinsic electron and hole concentrations are
(4-17)
ni  N c exp ( Ec  Ei ) / kT  , pi  N v exp  ( Ei  Ev ) / kT 
The product of n0 and p0 at equilibrium is a constant for a particular material and
temperature, even if the doping is varied:
n0 p0  ( N c exp ( Ec  E F ) / kT )( N v exp  ( E F  Ev ) / kT ) 

 N c N v exp  ( E c  E v ) / kT   N c N v exp  E g / kT

ni pi  ( N c exp  ( Ec  Ei ) / kT )( N v exp ( Ei  Ev ) / kT ) 

 N c N v exp  E g / kT
(4-18a)
(4-18b)

In Eqns. (4-18a) and (4-18b) Eg = Ec – Ev. The intrinsic electron and hole
concentrations are equal (since the carriers are created in pairs), ni = pi ; thus the
intrinsic concentration is
ni  N c N v exp(  Eg / 2kT )
(4-19)
The constant product of electron and hole concentrations in Eq. (4-18) can be
written conveniently as
n 0 p0  ni2
(4-20)
This is an important relation, and we shall use it extensively in later calculations. The
intrinsic concentration for Si at room temperature is approximately ni = 1.5 x 1010 cm-3.
Comparing Eqs. (4-17) and (4-19), we note that the intrinsic level Ei is the middle
of the band gap (Ec - Ei= Eg/2), if the effective densities of states Nc and Nv are equal.
There is usually some difference in effective mass for electrons and holes (e.g.
for Si – mn*=0.26m0, mn*=0.39m0), however, and, therefore, Nc and Nν are slightly
different as Eqs. (4-12) and (4-16) indicate.
Another convenient way of writing Eqs. (4-11) and (4-15) is
n0  ni exp ( EF  Ei ) / kT 
(4-21)
p0  ni exp ( Ei  EF ) / kT 
(4-22)
obtained by the application of Eq. (4-17). This form of the equations indicates
directly that the electron concentration is ni, when EF is at the intrinsic level Ei, and
that n0 increases exponentially as the Fermi level moves away from Ei toward the
conduction band. Similarly, the hole concentration p0 varies from ni, to larger values
as EF moves from Ei toward the valence band. Since these equations reveal the
qualitative features of carrier concentration so directly, they are particularly
convenient to remember.
Conductivity of Intrinsic and Extrinsic Semiconductors
  nq
n 0 p0  ni2
For Si n = 0.135 m2/Vs, p = 0.048 m2/Vs;
for Ge n = 0.39 m2/Vs, p = 0.19 m2/Vs.
Conductivity of Extrinsic Semiconductors
Typical carrier densities in intrinsic & extrinsic semiconductors
Si at 300K, intrinsic carrier density ni = 1.5 x 1016/m3
Extrinsic Si doped with As → typical concentration 1021atoms/m3:
Majority carriers n0 = 1021 e/m3 ; Mass action law: ni2 = n0p0
Minority carriers: p0 = (1.5×1016)2/1021 = 2.25 x 1011 holes/m3
Conductivity:
Majority carriers: n = 1021x0.135x1.6x10-19 (e/m3 ) (m2 /Vs) (As C) =0.216 ( cm)-1
Minority carriers: p = 2.25x10-11 x 0.048 x1.6x10-19 = 0.173x10-10 ( cm)-1
Conductivity total total = n + p  0.216 ( cm)-1
----------------------------------------------------------------------------------------------------------------------------
Conductivity of Intrinsic and Extrinsic Semiconductors:
Effect of Temperature
Illustrative Problem: calculate 
of Si at room temperature (20 oC
→293 K) and at 150 oC →423 K).
Increasing conductivity by doping
- Si - - Si -+- Si - Si - + - Si - - Si - - Si -+- Si - - Si - + - - +-+
B - - Si - - Si
B - Si - - Si - - Si
- + - - -+
B - - Si -+- Si - Si - - Si
- Si • Inject Boron into the crystal with an implant step.
• Boron is Group3 element is has 3 electrons in its outer shell (one less than silicon)
• This introduces holes into the lattice which can be made mobile by applying heat. This
gives us a hole current
• The result is a P-type semiconductor (p for positive current carrier)
Increasing conductivity by doping
-
-
Si
-
--
-
Si
-
Si
-+--
-
--Si - As
--Si - As
Si -
-
Si
-
Si
-
- + -+- Si - - Si - -- - Si Si
-Si - - Si - Si
- As
- - Si - - Si - - Si -+• Inject Arsenic into the crystal with an implant step.
- Si • Arsenic is Group5 element
- with 5 electrons in its outer shell, (one more than
silicon).
• This introduces extra electrons into the lattice which can be released through
the application of heat and so produces and electron current
• The result here is an N-type semiconductor (n for negative current carrier)
Summary
Intrinsic semiconductors
Doped semiconductors
n-type
n0  ni exp ( EF  Ei ) / kT 
n 0 p0  n
2
i
p-type
p0  ni exp ( Ei  EF ) / kT 
pn Junction
The interface separating the n and p
regions is referred to as the
metallurgical junction.
For simplicity we will consider a step
junction in which the doping
concentration is uniform in each
region and there is an abrupt change
in doping at the junction.
Initially there is a very large density
gradient in both the electron and
hole concentrations. Majority carrier
electrons in the n region will begin
diffusing into the p region and
majority carrier holes in the p
region will begin diffusing into the n
region. If we assume there are no
external connections to the
semiconductor, then this diffusion
process cannot continue
indefinitely.
Diffusion
Let us assume that we have two boxes- one
contains red air molecules while another one
contains blue molecules. This could be due to
appropriate types of pollution.
Let us join these 2 boxes together and
remove the wall between them.
Each type of molecules starts to
move to the region of their low
concentration due to the
concentration gradient in the middle.
Eventually there would be a homogeneous mixture
of two types of molecules.
pn Junction
This cannot occur in the case of the charged particles
in a p-n junction because of the development of space
charge and the electric field .
As electrons diffuse from the n region,
positively charged donor atoms are left
behind. Similarly, as holes diffuse from the p
region, they uncover negatively charged
acceptor atoms. These are minority carriers.
The net positive and negative charges in the
n and p regions induce an electric field in the
region near the metallurgical junction, in the
direction from the positive to the negative
charge, or from the n to the p region.
The net positively and negatively charged regions
are shown in Figure. These two regions are
referred to as the space charge region (SCR).
Essentially all electrons and holes are swept out of
the space charge region by the electric field. Since
the space charge region is depleted of any mobile
charge, this region is also referred to as the
depletion region
Density gradients still exist in the majority carrier concentrations at each edge of the space charge
region. This produce a "diffusion force" that acts the electrons and holes at the edges of the space
charge region. The electric field in the SCR produces another force on the electrons and holes
which is in the opposite direction to the diffusion force for each type of particle. In thermal equilibrium, the diffusion force and the E-field () force exactly balance each other.
pn Junction – built-in potential barrier
No applied voltage across pn-junction
The junction is in thermal equilibrium —
the Fermi energy level is constant
throughout the entire system. The
conduction and valence band energies
must bend as we go through the space
charge region, since the relative
position of the conduction and valence
bands with respect to the Fermi energy
changes between p and n regions.
Electrons in the conduction band of the n region see a potential barrier in trying to move
into the conduction band of the p region. This potential barrier is referred to as the built-in
potential barrier and is denoted by Vbi (or V0). The built-in potential barrier maintains
equilibrium between majority carrier electrons in the n region and minority carrier electrons
in the p region, and also between majority carrier holes in the p region and minority carrier
holes in the n region. The potential Vbi maintains equilibrium, so no current is produced by
this voltage.
The intrinsic Fermi level is equidistant from the conduction band edge through the junction,
thus the built-in potential barrier can be determined as the difference between the intrinsic
Fermi levels in the p and n regions.
pn Junction
An applied voltage bias V appears across the
transition region of the junction rather than in the
neutral n and p region. Of course, there will be some
voltage drop in the neutral material, if a current flows
through it. But in most p-n junction devices, the
length of each region is small compared with its area,
and the doping is usually moderate to heavy; thus the
resistance is small in each neutral region, and only a
small voltage drop can be maintained outside the
space charge (transition) region. V consider to be
positive when the external bias is positive on the p
side relative to the n side.
The electrostatic potential barrier at the junction is
lowered by a forward bias Vf from the equilibrium
contact potential V0 to the smaller value V0-Vf. This
lowering of the potential barrier occurs because a
forward bias (p positive with respect to n) raises the
electrostatic potential on the p side relative to the n
side. For a reverse bias (V=-Vr ) the opposite occurs;
the electrostatic potential of the p side is depressed
relative to the n side, and the potential barrier at the
junction becomes larger (V0 + Vr ).
The electric field within the transition region can be
deduced from the potential barrier. We notice that the
field decreases with forward bias, since the applied
electric field opposes the buid-in field. With reverse
bias the field at the junction is increased by the
applied field, which is in the same direction as the
equilibrium field.
Apply voltage/electric field
-
+V
- + P-type
- Si - - Si -+- Si - - Si +
- - -+
+-+
++
B - - Si - - Si
B - Si - - Si -+- Si
- + - - + +
-+
++
+
B +- - Si -+- Si - Si - - Si
- Si - N-type
- - Si - - Si -+- Si -- +
-+
Si - - Si -+- Si - - Si - Si - - As
+
--Si - - Si - - As
Si - - Si - - Si - As
- - -
-V
Modulators of conductivity
Just reviewed how conductivity of a semiconductor is affected by:
Temperature – Increasing temperature causes conductivity to
increase
• Dopants – Increasing the number of dopant atoms (implant dose)
cause conductivity to increase.
• Holes are slower than electrons therefore n-type material is
more conductive than p-type material.
• These parameters are in addition to those normally affecting
conducting material,
Cross sectional area 
Resistance 
Length 
Resistance 
Silicon Resistivity Versus Dopant Concentration
1021
Dopant Concentration (atoms/cm3)
1020
1019
1018
1017
n-type
1016
p-type
1015
1014
1013
10-3
10-2
10-1
100
101
102
103
Electrical Resistivity (ohm-cm)
Redrawn from VLSI Fabrication Principles, Silicon and Gallium Arsenide, John Wiley & Sons, Inc.
PN Junction: No electrical bias applied
Si
Si
B
Si
B
Si
Si
Si
As
Si
Si
B
Si
Si
As
Si
As
Si
Si
As
B
Si
Si
B
Si
Si
Si
Si
As
Si
Si
Si
B
Si
Si
Si
Si
Si
As
Si
P-type
N-type
Mostly B & free holes
Mostly As & free electrons
• Diffusion effects – The holes and electrons move from area of high
concentration to areas of low concentration.
• Holes & electrons annihilate each other to form an area depleted of free
charge. This is known as the depletion region and blocks any further flow of
charge carriers across the junction
Physics of the Depletion Region
• When n and p type material are placed in contact with each
other, the electrons diffuses into the p-type region in order to
equalise the Fermi levels.
• This loss of electrons from the n-type material leaves the
surface layer positively charged.
• Similarly the p-type material will have a negatively charged
surface layer.
• Thus an electric field is established which opposes the diffusion
of electrons when the Fermi levels are equal (dynamic
equilibrium is established)
Size of the depletion region
– This can be calculated solving Poisson’s Eqn. for the voltage
distribution across the layer.
– The resultant equation shows that
N d ln  N a l p
Where N is the dopant concentration and l the length on
the p and n type sides.
– The length is given as
2   0 V Na
1
ln 
e
Nd ( Na  Nd )
2   0 V Nd
1
lp 
e
Na ( Na  Nd )
PN Junction = Capacitor
P-type
[semiconductor]
N-type
[semiconductor]
D
p
N
Depletion region, barrier to free flow of current from P to N  insulator
Basically it forms parallel plate capacitor
Q
C
V
The capacitance per unit area of the junction can be defined as:
1
 Na Nd
1 2 
C  V 2   0 e
2
 Na  Nd




1
2
PN Junction = Diode
BUT – PN is no ordinary capacitor, actually a diode
Forward Bias: Shrink depletion region, current dragged through the barrier
+V
-V
D
p
N
Once the difficulty
of getting through
the depletion region
has been overcome,
current can rise with
applied voltage
(Ohm’s law)
Reverse Bias: Grow depletion region, current finds it more and more difficult to
get through the barrier
• Little current flows
because barrier too high
- -V
+V
+
D
p
N
• However increasing
voltage further  high
electric field
• Depletion region
eventually breaks down
 reverse current